Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 157 | 161 | theorem to_rightInverse (hf : AntilipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) :
LipschitzWith K g := by |
intro x y
have := hf (g x) (g y)
rwa [hg x, hg y] at this
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 236 | 237 | theorem comp_hom_eq_id (α : X ≅ Y) {f : Y ⟶ X} : f ≫ α.hom = 𝟙 Y ↔ f = α.inv := by |
rw [← eq_comp_inv, id_comp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 234 | 236 | theorem IsCompact.disjoint_nhdsSet_right {l : Filter X} (hs : IsCompact s) :
Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x) := by |
simpa only [disjoint_comm] using hs.disjoint_nhdsSet_left
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated... | Mathlib/MeasureTheory/Measure/Dirac.lean | 127 | 129 | theorem ae_dirac_eq [MeasurableSingletonClass α] (a : α) : ae (dirac a) = pure a := by |
ext s
simp [mem_ae_iff, imp_false]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.RingTheory.Trace
import Mathlib.FieldTheory.Finite.GaloisField
#align_import field_theory.finite.trace from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/Finite/Trace.lean | 25 | 32 | theorem trace_to_zmod_nondegenerate (F : Type*) [Field F] [Finite F]
[Algebra (ZMod (ringChar F)) F] {a : F} (ha : a ≠ 0) :
∃ b : F, Algebra.trace (ZMod (ringChar F)) F (a * b) ≠ 0 := by |
haveI : Fact (ringChar F).Prime := ⟨CharP.char_is_prime F _⟩
have htr := traceForm_nondegenerate (ZMod (ringChar F)) F a
simp_rw [Algebra.traceForm_apply] at htr
by_contra! hf
exact ha (htr hf)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 762 | 766 | theorem prod_apply (i) :
b.prod b' i = Sum.elim (LinearMap.inl R M M' ∘ b) (LinearMap.inr R M M' ∘ b') i := by |
ext <;> cases i <;>
simp only [prod_apply_inl_fst, Sum.elim_inl, LinearMap.inl_apply, prod_apply_inr_fst,
Sum.elim_inr, LinearMap.inr_apply, prod_apply_inl_snd, prod_apply_inr_snd, Function.comp]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 585 | 589 | theorem contDiffGroupoid_le (h : m ≤ n) : contDiffGroupoid n I ≤ contDiffGroupoid m I := by |
rw [contDiffGroupoid, contDiffGroupoid]
apply groupoid_of_pregroupoid_le
intro f s hfs
exact ContDiffOn.of_le hfs h
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 54 | 61 | theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 213 | 219 | theorem TendstoUniformlyOn.congr {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s)
(hff' : ∀ᶠ n in p, Set.EqOn (F n) (F' n) s) : TendstoUniformlyOn F' f p s := by |
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at hf ⊢
refine hf.congr ?_
rw [eventually_iff] at hff' ⊢
simp only [Set.EqOn] at hff'
simp only [mem_prod_principal, hff', mem_setOf_eq]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... | Mathlib/Data/Nat/Fib/Basic.lean | 135 | 143 | theorem le_fib_self {n : ℕ} (five_le_n : 5 ≤ n) : n ≤ fib n := by |
induction' five_le_n with n five_le_n IH
·-- 5 ≤ fib 5
rfl
· -- n + 1 ≤ fib (n + 1) for 5 ≤ n
rw [succ_le_iff]
calc
n ≤ fib n := IH
_ < fib (n + 1) := fib_lt_fib_succ (le_trans (by decide) five_le_n)
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import d... | Mathlib/Data/PNat/Factors.lean | 414 | 420 | theorem prod_sup (u v : PrimeMultiset) : (u ⊔ v).prod = PNat.lcm u.prod v.prod := by |
let n := u.prod
let m := v.prod
change (u ⊔ v).prod = PNat.lcm n m
have : u = n.factorMultiset := u.factorMultiset_prod.symm; rw [this]
have : v = m.factorMultiset := v.factorMultiset_prod.symm; rw [this]
rw [← PNat.factorMultiset_lcm n m, PNat.prod_factorMultiset]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 1,491 | 1,496 | theorem algEquiv_apply {x y : L} (hx : IsAlgebraic K x) (h_mp : minpoly K x = minpoly K y) :
algEquiv hx h_mp (AdjoinSimple.gen K x) = AdjoinSimple.gen K y := by |
have hy : IsAlgebraic K y := ⟨minpoly K x, ne_zero hx.isIntegral, (h_mp ▸ aeval _ _)⟩
rw [algEquiv, trans_apply, ← adjoinRootEquivAdjoin_apply_root K hx.isIntegral,
symm_apply_apply, trans_apply, AdjoinRoot.algEquivOfEq_apply_root,
adjoinRootEquivAdjoin_apply_root K hy.isIntegral]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 203 | 219 | theorem of_diff_of_diff_eq_zero {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
(h' : v (B \ A) = 0) : v (A \ B) + v B = v A := by |
symm
calc
v A = v (A \ B ∪ A ∩ B) := by simp only [Set.diff_union_inter]
_ = v (A \ B) + v (A ∩ B) := by
rw [of_union]
· rw [disjoint_comm]
exact Set.disjoint_of_subset_left A.inter_subset_right disjoint_sdiff_self_right
· exact hA.diff hB
· exact hA.inter hB
_ = v (A \ ... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 416 | 416 | theorem coeff_zero_mul_X (φ : R⟦X⟧) : coeff R 0 (φ * X) = 0 := by | simp
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 499 | 531 | theorem jacobiTheta₂'_functional_equation (z τ : ℂ) :
jacobiTheta₂' z τ = 1 / (-I * τ) ^ (1 / 2 : ℂ) * cexp (-π * I * z ^ 2 / τ) / τ *
(jacobiTheta₂' (z / τ) (-1 / τ) - 2 * π * I * z * jacobiTheta₂ (z / τ) (-1 / τ)) := by |
rcases le_or_lt (im τ) 0 with hτ | hτ
· rw [jacobiTheta₂'_undef z hτ, jacobiTheta₂'_undef, jacobiTheta₂_undef, mul_zero,
sub_zero, mul_zero] <;>
rw [neg_div, neg_im, one_div, inv_im, neg_nonpos] <;>
exact div_nonneg (neg_nonneg.mpr hτ) (normSq_nonneg τ)
have hτ' : 0 < (-1 / τ).im := by
rw [div_... |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 167 | 177 | theorem basisSets_smul_left (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets) :
∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y : E => x • y) ⁻¹' U := by |
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
rw [hU]
by_cases h : x ≠ 0
· rw [(s.sup p).smul_ball_preimage 0 r x h, smul_zero]
use (s.sup p).ball 0 (r / ‖x‖)
exact ⟨p.basisSets_mem s (div_pos hr (norm_pos_iff.mpr h)), Subset.rfl⟩
refine ⟨(s.sup p).ball 0 r, p.basisSets_mem s hr, ?_⟩
simp only [... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 121 | 124 | theorem integral_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) (hts : t ⊆ s) :
∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ - ∫ x in t, f x ∂μ := by |
rw [eq_sub_iff_add_eq, ← integral_union, diff_union_of_subset hts]
exacts [disjoint_sdiff_self_left, ht, hfs.mono_set diff_subset, hfs.mono_set hts]
|
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.AbsoluteValue
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.UniformSpace.C... | Mathlib/Topology/UniformSpace/CompareReals.lean | 60 | 65 | theorem Rat.uniformSpace_eq :
(AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by |
ext s
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
simp only [Rat.dist_eq, AbsoluteValue.abs_apply, ← Rat.cast_sub, ← Rat.cast_abs, Rat.cast_lt,
abs_sub_comm]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | Mathlib/Analysis/Calculus/MeanValue.lean | 728 | 738 | theorem exists_ratio_hasDerivAt_eq_ratio_slope :
∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by |
let h x := (g b - g a) * f x - (f b - f a) * g x
have hI : h a = h b := by simp only [h]; ring
let h' x := (g b - g a) * f' x - (f b - f a) * g' x
have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx =>
((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
have hhc : Continu... |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 402 | 402 | theorem cos_coe_pi : cos (π : Angle) = -1 := by | rw [cos_coe, Real.cos_pi]
|
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instance... | Mathlib/Algebra/AddConstMap/Basic.lean | 90 | 91 | theorem map_add_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (x : G) (n : ℕ) : f (x + n) = f x + n := by | simp
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e962... | Mathlib/Order/Height.lean | 352 | 366 | theorem wellFoundedGT_of_chainHeight_ne_top (s : Set α) (hs : s.chainHeight ≠ ⊤) :
WellFoundedGT s := by |
-- Porting note: added
haveI : IsTrans { x // x ∈ s } (↑· < ↑·) := inferInstance
obtain ⟨n, hn⟩ := WithTop.ne_top_iff_exists.1 hs
refine ⟨RelEmbedding.wellFounded_iff_no_descending_seq.2 ⟨fun f ↦ ?_⟩⟩
refine n.lt_succ_self.not_le (WithTop.coe_le_coe.1 <| hn.symm ▸ ?_)
refine le_iSup₂_of_le ((ofFn (n := n.... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Ring.Subring.Basic
#align_import field_theory.subfield from "leanprover-community/mathlib"@"28aa996fc6fb4317f008... | Mathlib/Algebra/Field/Subfield.lean | 781 | 789 | theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (· ≤ ·) S)
{x : K} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by |
let s : Subfield K :=
{ __ := Subring.copy _ _ (Subring.coe_iSup_of_directed hS).symm
inv_mem' := fun _ hx ↦ have ⟨i, hi⟩ := Set.mem_iUnion.mp hx
Set.mem_iUnion.mpr ⟨i, (S i).inv_mem hi⟩ }
have : iSup S = s := le_antisymm
(iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set K)) i) (Set.iUnion_subset ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 102 | 127 | theorem integral_comp_abs {f : ℝ → ℝ} :
∫ x, f |x| = 2 * ∫ x in Ioi (0:ℝ), f x := by |
have eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x := by
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [abs_eq_self.mpr (le_of_lt (by exact hx))]
by_cases hf : IntegrableOn (fun x => f |x|) (Ioi 0)
· have int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) := by
rw [← Measure... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 754 | 762 | theorem setToL1S_add (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
(h_add : FinMeasAdditive μ T) (f g : α →₁ₛ[μ] E) :
setToL1S T (f + g) = setToL1S T f + setToL1S T g := by |
simp_rw [setToL1S]
rw [← SimpleFunc.setToSimpleFunc_add T h_add (SimpleFunc.integrable f)
(SimpleFunc.integrable g)]
exact
SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _)
(add_toSimpleFunc f g)
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 2,390 | 2,395 | theorem mem_verts_toSubgraph (p : G.Walk u v) : w ∈ p.toSubgraph.verts ↔ w ∈ p.support := by |
induction' p with _ x y z h p' ih
· simp
· have : w = y ∨ w ∈ p'.support ↔ w ∈ p'.support :=
⟨by rintro (rfl | h) <;> simp [*], by simp (config := { contextual := true })⟩
simp [ih, or_assoc, this]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 87 | 89 | theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by |
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 804 | 806 | theorem isBigO_norm_left : (fun x => ‖f' x‖) =O[l] g ↔ f' =O[l] g := by |
simp only [IsBigO_def]
exact exists_congr fun _ => isBigOWith_norm_left
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 473 | 474 | theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by |
simp only [compl_iInter, compl_compl]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanpro... | Mathlib/Algebra/Polynomial/Splits.lean | 357 | 360 | theorem eq_prod_roots_of_monic_of_splits_id {p : K[X]} (m : Monic p)
(hsplit : Splits (RingHom.id K) p) : p = (p.roots.map fun a => X - C a).prod := by |
convert eq_prod_roots_of_splits_id hsplit
simp [m]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Ring.Divisibility.Basic
... | Mathlib/RingTheory/Prime.lean | 70 | 73 | theorem Prime.abs [LinearOrder α] {p : α} (hp : Prime p) : Prime (abs p) := by |
obtain h | h := abs_choice p <;> rw [h]
· exact hp
· exact hp.neg
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 756 | 758 | theorem zmodEquivZPowers_symm_apply_zpow (i : ℤ) :
h.zmodEquivZPowers.symm (Additive.ofMul (⟨ζ ^ i, i, rfl⟩ : Subgroup.zpowers ζ)) = i := by |
rw [← h.zmodEquivZPowers.symm_apply_apply i, zmodEquivZPowers_apply_coe_int]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSe... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 47 | 48 | theorem constantCoeff_invUnitsSub (u : Rˣ) : constantCoeff R (invUnitsSub u) = 1 /ₚ u := by |
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 60 | 64 | theorem hasseDeriv_apply :
hasseDeriv k f = f.sum fun i r => monomial (i - k) (↑(i.choose k) * r) := by |
dsimp [hasseDeriv]
congr; ext; congr
apply nsmul_eq_mul
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 346 | 353 | theorem sInf_eq_boundedBy_sInfGen (m : Set (OuterMeasure α)) :
sInf m = OuterMeasure.boundedBy (sInfGen m) := by |
refine le_antisymm ?_ ?_
· refine le_boundedBy.2 fun s => le_iInf₂ fun μ hμ => ?_
apply sInf_le hμ
· refine le_sInf ?_
intro μ hμ t
exact le_trans (boundedBy_le t) (iInf₂_le μ hμ)
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib... | Mathlib/Analysis/Calculus/LHopital.lean | 358 | 372 | theorem lhopital_zero_atBot (hff' : ∀ᶠ x in atBot, HasDerivAt f (f' x) x)
(hgg' : ∀ᶠ x in atBot, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in atBot, g' x ≠ 0)
(hfbot : Tendsto f atBot (𝓝 0)) (hgbot : Tendsto g atBot (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l := ... |
rw [eventually_iff_exists_mem] at *
rcases hff' with ⟨s₁, hs₁, hff'⟩
rcases hgg' with ⟨s₂, hs₂, hgg'⟩
rcases hg' with ⟨s₃, hs₃, hg'⟩
let s := s₁ ∩ s₂ ∩ s₃
have hs : s ∈ atBot := inter_mem (inter_mem hs₁ hs₂) hs₃
rw [mem_atBot_sets] at hs
rcases hs with ⟨l, hl⟩
have hl' : Iio l ⊆ s := fun x hx => hl x... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 536 | 557 | theorem adjugate_adjugate (A : Matrix n n α) (h : Fintype.card n ≠ 1) :
adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A := by |
-- get rid of the `- 2`
cases' h_card : Fintype.card n with n'
· haveI : IsEmpty n := Fintype.card_eq_zero_iff.mp h_card
apply Subsingleton.elim
cases n'
· exact (h h_card).elim
rw [← h_card]
-- express `A` as an evaluation of a polynomial in n^2 variables, and solve in the polynomial ring
-- where... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,487 | 1,494 | theorem integral_mono_measure {f : α → ℝ} {ν} (hle : μ ≤ ν) (hf : 0 ≤ᵐ[ν] f)
(hfi : Integrable f ν) : ∫ a, f a ∂μ ≤ ∫ a, f a ∂ν := by |
have hfi' : Integrable f μ := hfi.mono_measure hle
have hf' : 0 ≤ᵐ[μ] f := hle.absolutelyContinuous hf
rw [integral_eq_lintegral_of_nonneg_ae hf' hfi'.1, integral_eq_lintegral_of_nonneg_ae hf hfi.1,
ENNReal.toReal_le_toReal]
exacts [lintegral_mono' hle le_rfl, ((hasFiniteIntegral_iff_ofReal hf').1 hfi'.2).... |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 207 | 209 | theorem mem_top_objs (c : C) : c ∈ (⊤ : Subgroupoid C).objs := by |
dsimp [Top.top, objs]
simp only [univ_nonempty]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Ashvni Narayanan
-/
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Math... | Mathlib/NumberTheory/FunctionField.lean | 179 | 195 | theorem InftyValuation.map_add_le_max' (x y : RatFunc Fq) :
inftyValuationDef Fq (x + y) ≤ max (inftyValuationDef Fq x) (inftyValuationDef Fq y) := by |
by_cases hx : x = 0
· rw [hx, zero_add]
conv_rhs => rw [inftyValuationDef, if_pos (Eq.refl _)]
rw [max_eq_right (WithZero.zero_le (inftyValuationDef Fq y))]
· by_cases hy : y = 0
· rw [hy, add_zero]
conv_rhs => rw [max_comm, inftyValuationDef, if_pos (Eq.refl _)]
rw [max_eq_right (WithZer... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 496 | 516 | theorem oangle_add_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
o.oangle x (x + r • o.rotation (π / 2 : ℝ) x) = Real.arctan r := by |
rcases lt_trichotomy r 0 with (hr | rfl | hr)
· have ha : o.oangle x (r • o.rotation (π / 2 : ℝ) x) = -(π / 2 : ℝ) := by
rw [o.oangle_smul_right_of_neg _ _ hr, o.oangle_neg_right h, o.oangle_rotation_self_right h, ←
sub_eq_zero, add_comm, sub_neg_eq_add, ← Real.Angle.coe_add, ← Real.Angle.coe_add,
... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,347 | 1,356 | theorem integral_deriv_mul_eq_sub_of_hasDeriv_right {u v u' v' : ℝ → A}
(hu : ContinuousOn u [[a, b]])
(hv : ContinuousOn v [[a, b]])
(huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x)
(hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x)
(hu' : Interva... |
apply integral_eq_sub_of_hasDeriv_right (hu.mul hv) fun x hx ↦ (huu' x hx).mul (hvv' x hx)
exact (hu'.mul_continuousOn hv).add (hv'.continuousOn_mul hu)
|
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
import... | Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 533 | 534 | theorem modular_T_smul (z : ℍ) : ModularGroup.T • z = (1 : ℝ) +ᵥ z := by |
simpa only [Int.cast_one] using modular_T_zpow_smul z 1
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 545 | 551 | theorem Integrable.integral_eq_integral_meas_le
(f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t ≤ f a}) := by |
rw [Integrable.integral_eq_integral_meas_lt f_intble f_nn]
apply integral_congr_ae
filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f] with t ht
exact congrArg ENNReal.toReal ht.symm
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import ... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 108 | 118 | theorem IsFraction.eq_mk' {U : Opens (PrimeSpectrum.Top R)} {f : ∀ x : U, Localizations R x}
(hf : IsFraction f) :
∃ r s : R,
∀ x : U,
∃ hs : s ∉ x.1.asIdeal,
f x =
IsLocalization.mk' (Localization.AtPrime _) r
(⟨s, hs⟩ : (x : PrimeSpectrum.Top R).asIdeal.primeC... |
rcases hf with ⟨r, s, h⟩
refine ⟨r, s, fun x => ⟨(h x).1, (IsLocalization.mk'_eq_iff_eq_mul.mpr ?_).symm⟩⟩
exact (h x).2.symm
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 311 | 312 | theorem rank_fun' : Module.rank R (η → R) = Fintype.card η := by |
rw [rank_fun_eq_lift_mul, rank_self, Cardinal.lift_one, mul_one]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import ... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 400 | 403 | theorem ι_mul_ι (r₁ r₂) : ι (0 : QuadraticForm R R) r₁ * ι (0 : QuadraticForm R R) r₂ = 0 := by |
rw [← mul_one r₁, ← mul_one r₂, ← smul_eq_mul R, ← smul_eq_mul R, LinearMap.map_smul,
LinearMap.map_smul, smul_mul_smul, ι_sq_scalar, QuadraticForm.zero_apply, RingHom.map_zero,
smul_zero]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import ... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 101 | 107 | theorem embeddingPiTangent_ker_mfderiv (x : M) (hx : x ∈ s) :
LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) = ⊥ := by |
apply bot_unique
rw [← (mdifferentiable_chart I (f.c (f.ind x hx))).ker_mfderiv_eq_bot
(f.mem_chartAt_ind_source x hx),
← comp_embeddingPiTangent_mfderiv]
exact LinearMap.ker_le_ker_comp _ _
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 239 | 247 | theorem ae_eq_condexp_of_forall_setIntegral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStrongly... |
refine ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => ?_) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, setIntegral_condexp hm hf hs]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability... | Mathlib/Probability/Variance.lean | 143 | 143 | theorem evariance_zero : evariance 0 μ = 0 := by | simp [evariance]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.ModelTheory.Basic
#align_import model_theory.language_map from "leanprover-c... | Mathlib/ModelTheory/LanguageMap.lean | 320 | 332 | theorem Injective.isExpansionOn_default {ϕ : L →ᴸ L'}
[∀ (n) (f : L'.Functions n), Decidable (f ∈ Set.range fun f : L.Functions n => ϕ.onFunction f)]
[∀ (n) (r : L'.Relations n), Decidable (r ∈ Set.range fun r : L.Relations n => ϕ.onRelation r)]
(h : ϕ.Injective) (M : Type*) [Inhabited M] [L.Structure M] :
... |
letI := ϕ.defaultExpansion M
refine ⟨fun {n} f xs => ?_, fun {n} r xs => ?_⟩
· have hf : ϕ.onFunction f ∈ Set.range fun f : L.Functions n => ϕ.onFunction f := ⟨f, rfl⟩
refine (dif_pos hf).trans ?_
rw [h.onFunction hf.choose_spec]
· have hr : ϕ.onRelation r ∈ Set.range fun r : L.Relations n => ϕ.onRelat... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 206 | 211 | theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SigmaFinite ν]
[TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν := by |
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with x hx
exact
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Basic
#align_import algebra.covariant_and_contravariant from "leanprover-co... | Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 170 | 176 | theorem Group.covariant_swap_iff_contravariant_swap [Group N] :
Covariant N N (swap (· * ·)) r ↔ Contravariant N N (swap (· * ·)) r := by |
refine ⟨fun h a b c bc ↦ ?_, fun h a b c bc ↦ ?_⟩
· rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a]
exact h a⁻¹ bc
· rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a] at bc
exact h a⁻¹ bc
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 1,101 | 1,107 | theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
(hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t) :=
calc
s ∩ f ⁻¹' interior t ⊆ interior (s ∩ f ⁻¹' t) :=
interior_maximal (inter_subset_inter (Subset.refl _) (preim... | rw [interior_inter, hs.interior_eq]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 197 | 199 | theorem setIntegral_indicator (ht : MeasurableSet t) :
∫ x in s, t.indicator f x ∂μ = ∫ x in s ∩ t, f x ∂μ := by |
rw [integral_indicator ht, Measure.restrict_restrict ht, Set.inter_comm]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,836 | 1,843 | theorem le_inter (h₁ : s ≤ t) (h₂ : s ≤ u) : s ≤ t ∩ u := by |
revert s u; refine @(Multiset.induction_on t ?_ fun a t IH => ?_) <;> intros s u h₁ h₂
· simpa only [zero_inter, nonpos_iff_eq_zero] using h₁
by_cases h : a ∈ u
· rw [cons_inter_of_pos _ h, ← erase_le_iff_le_cons]
exact IH (erase_le_iff_le_cons.2 h₁) (erase_le_erase _ h₂)
· rw [cons_inter_of_neg _ h]
... |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,031 | 1,044 | theorem iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
(⨅ v : HeightOneSpectrum R,
Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) = ⊥ := by |
ext x
rw [Algebra.mem_iInf]
constructor
on_goal 1 => by_cases hR : IsField R
· rcases Function.bijective_iff_has_inverse.mp
(IsField.localization_map_bijective (Rₘ := K) (flip nonZeroDivisors.ne_zero rfl : 0 ∉ R⁰) hR)
with ⟨algebra_map_inv, _, algebra_map_right_inv⟩
exact fun _ => Algebra.mem... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 1,527 | 1,528 | theorem union_insert (a : α) (s t : Finset α) : s ∪ insert a t = insert a (s ∪ t) := by |
simp only [insert_eq, union_left_comm]
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 97 | 97 | theorem min_self (a : α) : min a a = a := by | simp [min_def]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 72 | 73 | theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by |
simp_rw [mem_iInter]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib... | Mathlib/Topology/UrysohnsLemma.lean | 251 | 252 | theorem lim_of_nmem_U (c : CU P) (x : X) (h : x ∉ c.U) : c.lim x = 1 := by |
simp only [CU.lim, approx_of_nmem_U c _ h, ciSup_const]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,212 | 1,217 | theorem image_inter' (hf : LeftInvOn f' f s) : f '' (s₁ ∩ s) = f' ⁻¹' s₁ ∩ f '' s := by |
apply Subset.antisymm
· rintro _ ⟨x, ⟨h₁, h⟩, rfl⟩
exact ⟨by rwa [mem_preimage, hf h], mem_image_of_mem _ h⟩
· rintro _ ⟨h₁, ⟨x, h, rfl⟩⟩
exact mem_image_of_mem _ ⟨by rwa [← hf h], h⟩
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,072 | 1,074 | theorem mem_map : b ∈ map f s ↔ f.symm b ∈ s := by |
rw [← f.symm_symm, ← symm_map, f.symm_symm]
rfl
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 236 | 240 | theorem MDifferentiableWithinAt.hasMFDerivWithinAt (h : MDifferentiableWithinAt I I' f s x) :
HasMFDerivWithinAt I I' f s x (mfderivWithin I I' f s x) := by |
refine ⟨h.1, ?_⟩
simp only [mfderivWithin, h, if_pos, mfld_simps]
exact DifferentiableWithinAt.hasFDerivWithinAt h.2
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.BoundedOrder
#align_import algebra.order.mo... | Mathlib/Algebra/Order/Monoid/Defs.lean | 185 | 185 | theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 := by | simp [← not_iff_not]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 1,387 | 1,388 | theorem sep_eq_self_iff_mem_true : { x ∈ s | p x } = s ↔ ∀ x ∈ s, p x := by |
simp_rw [ext_iff, mem_sep_iff, and_iff_left_iff_imp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 1,398 | 1,399 | theorem single_smul (a b : α) (f : α → M) (r : R) : single a r b • f a = single a (r • f b) b := by |
by_cases h : a = b <;> simp [h]
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 484 | 486 | theorem Finite.infsep_of_nontrivial (hsf : s.Finite) (hs : s.Nontrivial) :
s.infsep = hsf.offDiag.toFinset.inf' (by simpa) (uncurry dist) := by |
classical simp_rw [hsf.infsep, dif_pos hs]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theor... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 210 | 225 | theorem mul_normal (H N : Subgroup G) [hN : N.Normal] : (↑(H ⊔ N) : Set G) = H * N := by |
rw [sup_eq_closure_mul]
refine Set.Subset.antisymm (fun x hx => ?_) subset_closure
induction hx using closure_induction'' with
| one => exact ⟨1, one_mem _, 1, one_mem _, mul_one 1⟩
| mem _ hx => exact hx
| inv_mem x hx =>
obtain ⟨x, hx, y, hy, rfl⟩ := hx
simpa only [mul_inv_rev, mul_assoc, inv_inv... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 356 | 373 | theorem partiallyWellOrderedOn_iff_finite_antichains [IsSymm α r] :
s.PartiallyWellOrderedOn r ↔ ∀ t, t ⊆ s → IsAntichain r t → t.Finite := by |
refine ⟨fun h t ht hrt => hrt.finite_of_partiallyWellOrderedOn (h.mono ht), ?_⟩
rintro hs f hf
by_contra! H
refine infinite_range_of_injective (fun m n hmn => ?_) (hs _ (range_subset_iff.2 hf) ?_)
· obtain h | h | h := lt_trichotomy m n
· refine (H _ _ h ?_).elim
rw [hmn]
exact refl _
· e... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 934 | 935 | theorem univ_pi_subset_univ_pi_iff :
pi univ t₁ ⊆ pi univ t₂ ↔ (∀ i, t₁ i ⊆ t₂ i) ∨ ∃ i, t₁ i = ∅ := by | simp [pi_subset_pi_iff]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 293 | 294 | theorem liftHom_root (h : IsAdjoinRoot S f) : h.liftHom x hx' h.root = x := by |
rw [← lift_algebraMap_apply, lift_root]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 2,266 | 2,273 | theorem normal_exists_closure_subset [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsOpen t)
(hst : s ⊆ t) : ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t := by |
have : Disjoint s tᶜ := Set.disjoint_left.mpr fun x hxs hxt => hxt (hst hxs)
rcases normal_separation hs (isClosed_compl_iff.2 ht) this with
⟨s', t', hs', ht', hss', htt', hs't'⟩
refine ⟨s', hs', hss', Subset.trans (closure_minimal ?_ (isClosed_compl_iff.2 ht'))
(compl_subset_comm.1 htt')⟩
exact fun x ... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 404 | 409 | theorem nullHomotopicMap_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.Rel k₀ l) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₀ = hom k₀ k₁ ≫ D.d k₁ k₀ := by |
dsimp only [nullHomotopicMap]
rw [prevD_eq hom r₁₀, dNext, AddMonoidHom.mk'_apply, C.shape, zero_comp, zero_add]
exact hk₀ _
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,017 | 1,024 | theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a)
(hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by |
apply le_antisymm
· apply PartENat.le_of_lt_add_one
rw [← Nat.cast_one, ← Nat.cast_add, lt_iff_not_ge, ge_iff_le,
le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]
simp
rw [le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]
|
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Yury G. Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Lemmas abou... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 88 | 92 | theorem pow_lt_pow_right' [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ} (ha : 1 < a)
(h : n < m) : a ^ n < a ^ m := by |
rcases Nat.le.dest h with ⟨k, rfl⟩; clear h
rw [pow_add, pow_succ, mul_assoc, ← pow_succ']
exact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 1,062 | 1,063 | theorem Filter.EventuallyEq.fderiv_eq (h : f₁ =ᶠ[𝓝 x] f) : fderiv 𝕜 f₁ x = fderiv 𝕜 f x := by |
rw [← fderivWithin_univ, ← fderivWithin_univ, h.fderivWithin_eq_nhds]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 347 | 350 | theorem mk_smul_mk (r : R) (m : M) (s t : S) :
Localization.mk r s • mk m t = mk (r • m) (s * t) := by |
rw [Localization.mk_eq_mk']
exact mk'_smul_mk ..
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 1,043 | 1,048 | theorem _root_.ContinuousOn.angle_sign_comp {α : Type*} [TopologicalSpace α] {f : α → Angle}
{s : Set α} (hf : ContinuousOn f s) (hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ π) :
ContinuousOn (sign ∘ f) s := by |
refine (ContinuousAt.continuousOn fun θ hθ => ?_).comp hf (Set.mapsTo_image f s)
obtain ⟨z, hz, rfl⟩ := hθ
exact continuousAt_sign (hs _ hz).1 (hs _ hz).2
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 1,030 | 1,031 | theorem fst_apply {s : Set α} (hs : MeasurableSet s) : ρ.fst s = ρ (Prod.fst ⁻¹' s) := by |
rw [fst, Measure.map_apply measurable_fst hs]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#a... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 220 | 224 | theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ := by |
refine funext fun x => ?_
rw [Complex.exp, exp_eq_tsum_div]
have : CauSeq.IsComplete ℂ norm := Complex.instIsComplete
exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable ℝ x).hasSum.tendsto_sum_nat
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 850 | 857 | theorem orthonormalBasis_one_dim (b : OrthonormalBasis ι ℝ ℝ) :
(⇑b = fun _ => (1 : ℝ)) ∨ ⇑b = fun _ => (-1 : ℝ) := by |
have : Unique ι := b.toBasis.unique
have : b default = 1 ∨ b default = -1 := by
have : ‖b default‖ = 1 := b.orthonormal.1 _
rwa [Real.norm_eq_abs, abs_eq (zero_le_one' ℝ)] at this
rw [eq_const_of_unique b]
refine this.imp ?_ ?_ <;> (intro; ext; simp [*])
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 93 | 99 | theorem braiding_tensor_left (X Y Z : C) :
(β_ (X ⊗ Y) Z).hom =
(α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
(β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom := by |
apply (cancel_epi (α_ X Y Z).inv).1
apply (cancel_mono (α_ Z X Y).inv).1
simp [hexagon_reverse]
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Alge... | Mathlib/Data/Nat/Cast/Order.lean | 142 | 143 | theorem cast_lt_one : (n : α) < 1 ↔ n = 0 := by |
rw [← cast_one, cast_lt, Nat.lt_succ_iff, ← bot_eq_zero, le_bot_iff]
|
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import... | Mathlib/SetTheory/Surreal/Dyadic.lean | 204 | 209 | theorem zsmul_pow_two_powHalf (m : ℤ) (n k : ℕ) :
(m * 2 ^ n) • powHalf (n + k) = m • powHalf k := by |
rw [mul_zsmul]
congr
norm_cast
exact nsmul_pow_two_powHalf' n k
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.RingTheory.HahnSeries.Addition
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
#align_impor... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 237 | 263 | theorem single_mul_coeff_add [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
{b : Γ} : (single b r * x).coeff (a + b) = r * x.coeff a := by |
by_cases hr : r = 0
· simp [hr, mul_coeff]
simp only [hr, smul_coeff, mul_coeff, support_single_of_ne, Ne, not_false_iff, smul_eq_mul]
by_cases hx : x.coeff a = 0
· simp only [hx, mul_zero]
rw [sum_congr _ fun _ _ => rfl, sum_empty]
ext ⟨a1, a2⟩
simp only [not_mem_empty, not_and, Set.mem_singleto... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 463 | 471 | theorem IsPreconnected.prod [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ×ˢ t) := by |
apply isPreconnected_of_forall_pair
rintro ⟨a₁, b₁⟩ ⟨ha₁, hb₁⟩ ⟨a₂, b₂⟩ ⟨ha₂, hb₂⟩
refine ⟨Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s, ?_, .inl ⟨b₁, hb₁, rfl⟩, .inr ⟨a₂, ha₂, rfl⟩, ?_⟩
· rintro _ (⟨y, hy, rfl⟩ | ⟨x, hx, rfl⟩)
exacts [⟨ha₁, hy⟩, ⟨hx, hb₂⟩]
· exact (ht.image _ (Continuous.Prod.mk _).continuous... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c73... | Mathlib/Topology/Maps.lean | 422 | 425 | theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
f ⁻¹' frontier s = frontier (f ⁻¹' s) := by |
simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,
hf.preimage_closure_eq_closure_preimage hfc]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 278 | 282 | theorem SameCycle.exists_pow_eq_of_mem_support (h : SameCycle f x y) (hx : x ∈ f.support) :
∃ i < (f.cycleOf x).support.card, (f ^ i) x = y := by |
rw [mem_support] at hx
exact Equiv.Perm.IsCycleOn.exists_pow_eq (b := y) (f.isCycleOn_support_cycleOf x)
(by rw [mem_support_cycleOf_iff' hx]) (by rwa [mem_support_cycleOf_iff' hx])
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 1,766 | 1,781 | theorem trTape'_move_left (L R : ListBlank Γ) :
(Tape.move Dir.left)^[n] (trTape' enc0 L R) = trTape' enc0 L.tail (R.cons L.head) := by |
obtain ⟨a, L, rfl⟩ := L.exists_cons
simp only [trTape', ListBlank.cons_bind, ListBlank.head_cons, ListBlank.tail_cons]
suffices ∀ {L' R' l₁ l₂} (_ : Vector.toList (enc a) = List.reverseAux l₁ l₂),
(Tape.move Dir.left)^[l₁.length]
(Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =
Ta... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 522 | 527 | theorem _root_.Basis.coe_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
(v.toOrthonormalBasis hv : ι → E) = (v : ι → E) :=
calc
(v.toOrthonormalBasis hv : ι → E) = ((v.toOrthonormalBasis hv).toBasis : ι → E) := by |
classical rw [OrthonormalBasis.coe_toBasis]
_ = (v : ι → E) := by simp
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 190 | 203 | theorem closure_inverse_graph (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
(hcf : LinearMap.ker f.closure.toFun = ⊥) :
f.closure.inverse.graph = f.inverse.graph.topologicalClosure := by |
rw [inverse_graph hf, inverse_graph hcf, ← hf'.graph_closure_eq_closure_graph]
apply SetLike.ext'
simp only [Submodule.topologicalClosure_coe, Submodule.map_coe, LinearEquiv.prodComm_apply]
apply (image_closure_subset_closure_image continuous_swap).antisymm
have h1 := Set.image_equiv_eq_preimage_symm f.graph... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Perm/Sign.lean | 602 | 604 | theorem sign_subtypeCongr {p : α → Prop} [DecidablePred p] (ep : Perm { a // p a })
(en : Perm { a // ¬p a }) : sign (ep.subtypeCongr en) = sign ep * sign en := by |
simp [subtypeCongr]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 366 | 368 | theorem norm_eq_norm (x : K) :
mixedEmbedding.norm (mixedEmbedding K x) = |Algebra.norm ℚ x| := by |
simp_rw [mixedEmbedding.norm_apply, normAtPlace_apply, prod_eq_abs_norm]
|
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 129 | 132 | theorem nat_card : Nat.card (DihedralGroup n) = 2 * n := by |
cases n
· rw [Nat.card_eq_zero_of_infinite]
· rw [Nat.card_eq_fintype_card, card]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fi... | Mathlib/Data/Fin/Tuple/Sort.lean | 205 | 208 | theorem antitone_pair_of_not_sorted' (h : f ∘ σ ≠ f ∘ sort f) :
∃ i j, i < j ∧ (f ∘ σ) j < (f ∘ σ) i := by |
contrapose! h
exact comp_sort_eq_comp_iff_monotone.mpr (monotone_iff_forall_lt.mpr h)
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
#align_impo... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 46 | 57 | theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s : Set ℂ} {c : ℂ}
(hs : s ∈ 𝓝 c) :
DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s := by |
refine ⟨?_, fun hd => ⟨hd.mono diff_subset, (hd.differentiableAt hs).continuousAt⟩⟩
rintro ⟨hd, hc⟩ x hx
rcases eq_or_ne x c with (rfl | hne)
· refine (analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
?_ hc).differentiableAt.differentiableWithinAt
refine eventually_nhdsWithin_iff.2 ((ev... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 214 | 214 | theorem span_one : span (1 : Set α) = ⊤ := by | rw [← Set.singleton_one, span_singleton_one]
|
/-
Copyright (c) 2021 Alex Kontorovich, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Homeomorph
import Mathlib.GroupTheory.GroupAction.Basic
im... | Mathlib/Topology/Algebra/ConstMulAction.lean | 389 | 394 | theorem isClosedMap_smul₀ {E : Type*} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E]
[T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) : IsClosedMap fun x : E => c • x := by |
rcases eq_or_ne c 0 with (rfl | hne)
· simp only [zero_smul]
exact isClosedMap_const
· exact (Homeomorph.smulOfNeZero c hne).isClosedMap
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 60 | 62 | theorem count_eq_card_fintype (n : ℕ) : count p n = Fintype.card { k : ℕ // k < n ∧ p k } := by |
rw [count_eq_card_filter_range, ← Fintype.card_ofFinset, ← CountSet.fintype]
rfl
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.